{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p24: Pseudospectra of Davies's complex harmonic oscillator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "from chebPy import cheb\n",
    "from numpy import dot,argsort,zeros,real,imag,meshgrid,eye,diag,arange\n",
    "from scipy.linalg import solve,eig,svd,svdvals\n",
    "from matplotlib.pyplot import figure,plot,title,axis,contour"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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   "source": [
    "N = 70; D, x = cheb(N); x = x[1:N];\n",
    "L = 6.0; x = L*x; D = D/L;\n",
    "A = -dot(D,D);\n",
    "A = A[1:N,1:N] + (1+3j)*diag(x**2);\n",
    "lam, v = eig(A)\n",
    "fig = figure()\n",
    "plot(real(lam),imag(lam),\"o\")\n",
    "axis([0, 50, 0, 40])\n",
    "\n",
    "h = 0.5 #Smaller the value, finer the plot\n",
    "x = arange(0,50+h,h); y = arange(0,40+h,h); xx,yy = meshgrid(x,y);\n",
    "zz = xx + 1j*yy;\n",
    "I = eye(N-1); sigmin = zeros((len(y),len(x))) \n",
    "for j in range(0,len(x)):\n",
    "    for i in range(0,len(y)):\n",
    "        sigmin[i,j] = min(svdvals(zz[i,j]*I - A));\n",
    "\n",
    "levels = 10.0**arange(-4.5,0.0,0.5);\n",
    "contour(x,y,sigmin,levels,colors = 'k');"
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